mapping class group of the Klein bottle


the Klein bottle is gotten from a quotient in the two-torus S^1\times S^1 via

K=\frac{S^1\times S^1}{(z,w)\sim(-z,\bar{w})}

so to get all the isotopy classes [f]:K\to K one need to consider the auto-homeomorphisms \phi:T\to T of the torus.

It is well known that each of those are determined by a two-by-two matrix of GL_2(\mathbb{Z}), i.e (z,w)\to (z^aw^b,z^cw^d) where \det\left(\begin{array}{cc}a&b\\c&d\end{array}\right)=ad-bc=\pm1, so in seeking those matrices which obey the above gluing conditions we are compeled to analyse

(-z)^a\bar{w}^b=-z^aw^b

(-z)^c\bar{w}^d=\bar{z}^c\bar{w}^d

for each couple z,w\in S^1.

Then if we set w=1 this implies (-1)^az^a=-z^a and (-1)^a=-1, so a is odd. Also,  if z=1 then \bar{w}^b=w^b and hence b=0 .

But ad=\pm1, then \left(\begin{array}{cc}\pm1&0\\ 0&\pm1\end{array}\right) are the only matrices which preserve the gluing conditions.

It is easy to see that these four matrices form the famous 4-Klein group. So {\cal{MCG}}(K)=\mathbb{Z}_2\oplus\mathbb{Z}_2.

 

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Filed under algebra, topology

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